Math 154 — Rodriguez
Angel — 10.2
Solving Equations by Completing the Square
I. Completing the Square
A. Methods to solve quadratic equations:
 Zero-Product Rule: if poly factors easily, then write equation as poly=0, factor poly,
set each factor = 0
 Square Root Property: if there is no middle term in poly or looks like (binomial)2 = #
then use x2 = #, then x = ± # .
B. We can “fix” quadratic equations so we can use the Square Root Property. We “fix”
them by rewriting them so they fit the form (binomial)2 = #. The process of rewriting is
called “completing the square” because we are writing one side as a perfect square.
Examples of perfect squares:
A 2 ! 2AB + B 2 = ( A ! B )
x 2 ! 6x + 9 = ( x ! 3)
2
A 2 + 2AB + B 2 = ( A + B )
2
x 2 + 10x + 25 = ( x + 5 )
2
2
What do we notice?
Examples: Complete the square.
1)
x 2 ! 14x + _____ =
3)
x 2 ! 12x + _____ =
2)
x 2 + 18x + _____ =
II. Solving Quadratic Equations by Completing the Square
Steps (when the poly’s leading term is x2):
1. Write the equation so that the squared term (x2) and middle term (bx term) are
on one side of the equation and the constant is on the other side.
2. Complete the square by taking _________ the coefficient of the middle term,
______________ and __________ that amount to ______ sides of the equation.
3. Rewrite the trinomial as a perfect square binomial. When done the equation will
look like (binomial)2 = #.
4. Apply the Square Root Property.
5. Solve the resulting equation. Simplify answers.
Examples: Solve by completing the square.
1) x 2 ! 4x ! 6 = 0
2) x 2 + 2 = !6x
3) x 2 + 2x ! 5 = 0
4) x 2 ! 8x = 10